The Problem

The Watchman's Walk Problem and its Variations

by

©

Rebecca K eping

A thesi submitted to the School of Gra luate Studies

in partial fulfilment of the requir mcnts for the degree of

l\Iaster of cience

Department of Mathemati s and Statisti s Memorial Univ rsity of Newfoundland

N ovcmbcr 2009

t. John's Newfoundland

Abstract

Given a graph and a single watchman, the Watchman

's Walk Problem is concerned

with finding closed dominating walks of minimum length, which the watchman

can

traverse to efficiently guard the graph. When multiple guards are available, two nat- ural variations emerg :

(

1)

given a

fixed numb

er of guards,

how can we minimize

th

length oftime for which vertices are unobserved? and

(2) given fixed time constraints

on the monitoring of vertices, what is the minimum number of guards requir d? The present thesis reviews known results for the original problem as well

as its variations,

and proves an upp er bound on the number

of guards required when time is fix d.

Acknowledgements

I would like to express a great deal of gratitude to my supervisors Dr. Danny Dyer and Dr. Chris Radford. The timely completion of this thesis would not have b en possible without Dr. Dyer's dedicated guidance. I would also like to thank my family and friends for their continued encouragement, and Johnathan for his support, patienc , and counsel.

Contents

Abstract

Acknowledgements

List of Figures

1 Introduction

1.1

Motivation .

1. 2 Definitions .

1.3 Variations on the

problem

2 Fixed number of guards

2.1 One guard:

the original watchman problem . 2. 2 Mult

ipl guards:

downsizing a dominating set

3 Fixed time

3.1

3.2

3.3

Introductory results .

.

.

.

. . . . .

. A generalized upp

er

bound for

odd t

Analysis of the bound

11

iii

Vl

1

1

4 6

9 9

19

27 27

33

40

4 Bounds for small even t 45

5 Conclusions and open questions 54

Bibliography 57

List of Figures

1.1 A minimum dominating set (shaded, left), closed dominating walk (centre), and MCDW (right) of a graph

G. . . . . . . . . . . . . . .

3

2.1 A closed dominating walk for

G

obtained from the spanning tree

T.

12 2.2

A

graph satisfying

w

₁

(G)

= 2IE(To)l that is not a tree. . . . . 13 2.3 Three choices for a spanning tree

T ,

and the resulting dominating walks. 14 2.4 A graph of girth 4 for which w₁ (G)

<

2IE(T_{0 )} I for any spanning tree

T.

15 2.5 A cactus G, the graphs G' and G" from Theorem 2.11, and a MCDW. 19 2.6 Different methods of monitoring a graph with two guards (g1 and g2). 21

3.1

A

graph G that is 2-monitored with four guards. . . . . . . . . 28 3.2 The length of time for which each vertex in G has been unobserved. 29 3.3 TI:·ees satisfying W₁

(T) = l n2

¹

j . . . .

3.4 A tree T with W₃(T)

=

3

> l n+z-

⁴

j.

31 42 3.5

A

tre T satisfying

vVt

^(T)

⁼ l n+z-

³

^j

fort odd and k

=

t~³, 2 ::; j ::; k+ 1. 44

4.1 Possible subtrees Si when

t =

2 . . . . 4.2 Choices for the subtree

T'

when t

=

4 (Case 1).

4.3 Choices for the subtree

T'

when t

=

4 (Case 2).

48 49 52

Chapter 1

Introduction

1.1 Motivation

A museum

is att

mpting

to

monitor its rooms. Each room is connec ted

to one or

more

other

rooms via hallway , and from

any given

room it i

possible to see all adjacent room . Placing one guard in every room will en

ure all rooms arc constantly monitor d

, but thi

r quires more guards

than are

necessary:

we

need

only pla guards iu ~uch a

way

LhaL every

room

ei.Lher

h

as a guard r

is adj

acent to a

r

o

m with a guard. This problem belong

to the field of graph the

ry, and the set of room

we

requir i · called

a dominating set.

In graph theory, a graph G

is a et

V (G) of vertices together with a set E (G) of edges.

Th

-

dges

of G are subsets of size two

from

V(G), a

nd

we ay two vertices

'U, v E V (G) arc adjacent or neighbou1·ing

if the edge {

u, v} (usually written uv)

belong

to

E(G). More generally, if

we allow for

mult iple edge

bctw en the same

pair of vertice · t hen we

obtain a multigraph, and

if we

a cept edges of the form uu

called loops, we obtain a reflexive graph. Here, however, the term graph will be restricted to what is sometimes called a simple graph: a graph with no multiple edges and no loops.

It is not difficult to see how the museum problem can be translated into the language of graph theory. The museum is a graph, say G, with rooms as vertices and halls as edges. The definition of a dominating set is then a set D ~ V(G) with the property that every vertex of G is either in D or adjacent to a vertex of D. The concept of graph domination is widely researched, and many results are known about the domination number of a graph: the size of a smallest dominating s t, d noted 1(

G) .

Minimizing the size of a dominating set is important, as 'wasteful' dominating sets ar a ·y to find (take D

=

V (G), for example).

A variation on domination, as introduced by Hartnell, Rall, and Whitehead in 1998 [4], consider an alt rnative method of guarding the museum: rather than placing one guard in each room of a dominating set, have a single guard (or 'watchman') walk around the museum in such a way that the visited rooms collectively form a dominating s t. This ensures that every room has been either visited by the guard or seen by the guard from an adjacent room. We will assume that the guard's route begins and ends in the same room, allowing the walk to be repeated.

In a graph, an alternating sequence of vertices and edges, such as the route of a guard through a museum, is called a walk; more formally, a walk of length k is a sequence _{v 0,} e₁, v1 , e 2 , ... , ek, vk where ei

=

^vi-lvi for each i. Note that the length of a walk is the number of edges it contains. A walk is closed if it begins and ends on the same vertex and is dominating if the vertices of th walk form a dominating set.

We see then that the desired route for a single museum guard as described above is

a closed dominating walk.

The

added economic efficiency of this method, as only one guard is r quir d for the whole

museum

, is gained at the sacrifice

of security, since at

any given time th

r will b

e room t

hat

are not visible by the guard.

We will therefore be concerne l

again with minimality;

in particular, we wa nt to find

a shortest

route for

the guard to

walk. This

is the

Watchman's Walk Problem:

given a graph G, find a

dominating walk

that

is closed

and of minimum length, or a minimum closed dominating walk (MCDW) in G.

We will use

w₁ (G)

to denote the length

of a

MCDW in

a graph

G, wh re

the 1

indicates

that a single guard is

walking G.

Although

a closed dominating walk can be construct d from a dominating set D

by forming

an alternating sequence of vertices and edges that at l ast

include

all

vertices of D ,

thi is

not generally the most ffective method

, even if

D is minimum.

Figure

1.1

illustrates this point;

in

fact

, from the graph G we see that a

MCDW need not even contain a minimum dominating set. The watchman

's walk is t

hus a distinctly differ

ent problem from that of finding a minimum dominating set in a graph. MCDWs are

further

xplored

in

Chapter

2. Before we introduce

the primary objective of th

present thesis, a more thorough introduction to graph theory is required

; the following section provid s t

he

necessary

background terminology.

G G G

Figure 1.1:

A minimum dominating set

(shaded, left), closed

dominating walk

( cen- tre), and MCDW (right) of a graph

G.

1.2 D e finition s

Th number of vertices IV(G)I

in a graph G

is called the

order of G, a

nd the number of edges IE(

G)

I

is called

the

size

of

G. If

vertices

u and v are adjacent we say u

is

a neighbour

of

v, and

the

set

of

all

neighbours of

v along with v itself

is called

the closed neighbo'urhood

of

'U,

denoted N[v].

If e

=

^'UV is the edge joining u and v then we

say

u and v are

both

incident

with

e.

The number

of edges incident with the vertex v

in a

graph G is called the degree

of

v and is

denoted

degcv, or simply deg v

if th

e

associated graph is clear from

context.

A vertex of degree 0 is

called an isolate.

A

graph

with

n

vertices,

every

two of which

are adjacent,

is

call d the complete graph

of order

n,

denoted

Kn-

A

bipartite graph

is

one

whose v r

tices can be palti-

tioned

into

two

sets

A

and

B

such

that

every edge in the graph has

one end in A

and the other in B; similarly, a multipartite gmph

has its vertex

set

partitioned into multiple sets such that no vertex has a

neighbour in its own set. The term 'complet ' is applied to a

bipartite or multipartit

e graph when all

possible

dges are

present. A

complete bipartite graph

that has one set of size 1 and the others

t

of size k is called

a k-star.

The concept of a

walk in a graph, as introduced in Section 1.1

, leads to a

number

of

further

definitions. For

example, a u-v walk is a

walk beginning

on the vertex u and ending

on the vertex

v.

A closed walk with

all edges

distinct

is called a circuit, and a walk with both vertices and edges distinct is called a path.

A closed

circuit with

no

repeated

vertices

except

for the first

and

last is

called a cycle; a cycle of length k is called a

k-cycle

and

is

d noted

Ck.

If a circuit

in a graph

G visits every edge of G exactly

once then

it

is

called

an

Eulerian circuit, and

when such

a

walk

exists G

- - - -- - - -

is said to be Eulerian. It is a well-known result, originally observed by Euler, that a graph is Eul rian if and only if each of its vertices has even degree. A Hamilton cycle in a graph G is a cycle which includes every vertex of G exactly once, and if such a cycle exists then th graph G is said to be Hamiltonian.

A graph His a subgraph of a graph G if V(H) ~

V(G)

and E(H) ~ E(G). A spanning subgraph of G is a subgraph of G with vertex set V (G). An induced subgraph of G is a subgraph H whose edge set E(H) consists of all edges of G that have both endpoints in V (H). For a set of vertices S ~ V (G) we use G \ S (or G \ v if S contains a single vertex

v)

to denot the induced sub graph with vertex set

V

(G) \

S,

and for a set of edges S ~ E(G) we denote by G \ S the subgraph with vertex set V(G) and edge set E(G) \ S.

A graph is said to be connected if there is a path between any two vertices, and the maximal connected subgraphs of a disconnect d graph are the components of the graph. If the graph G is connected and the graph G \ v is disconnected then the vertex v E V (G) is called a cut vertex; similarly, an edge whose removal disconnects the graph is called a cut edge. A maximal connected subgraph containing no cut vertices is called a block of the underlying graph. A cactus i a graph with the prop rty that each of its blocks is either an edge or a cycle.

The girth of a graph is the length of a shortest cycle in the graph. The girth of a graph that contains no cycles is defined to be infinity. An important family of graphs called trees are categorized by the abs nee of cycles; equivalently, a tree is a graph which has a unique u-v path for any two vertices u, v. ote that this definition forces trees to be connected. A spanning tr·ee of a graph G is a spanning subgraph of G that is a tree. Vve refer to vertices of degree 1 in a tree as leaves and to a leaf's single

neighbouring vertex as a stem.

If T is a tree

then

L(T)

denotes the set of leaves of

T, and we

will define T

0

to be t

he leaf-deleted subtree T \ L(T).

If a graph G has a u-v path

then t

he distance in G from u to v,

written

dc(u, v)

(or

d(u, v)

when G

is

clear)

, is

the

length

of a shor test u-v path in

G. If S

is

a

set of vertices

in

G then the distance from

a

vertex

v (j.

S to

the

set S

is given

by

d(v, S) = min d(u, v).

uES

We

move

now fr

om basic background terminology to a few sp

ecific concepts that

will appear in forthcoming discussions: matchings and pair

ed

dominat ion. A matching

in

a gr

aph G is

a

set

of edges of G

t

hat hav no

common endpoints.

A

maximum matching is one containing

the greatest number of edges, and

a perfect matching is

one which

uses every vertex

of the

graph. A total dominating set of a graph G is a

dominating set of G

with the

property that

every

vertex

of G

ha · a neighbour

in D.

At first this may

not appear to

be different from the original definition

of a dominating set; however, the set

of shaded vertices in F

igure

1.1

is an exam

pl

of a

dominating set t

hat is not a total dominating set, since

neither of the shaded

vertices

has a

neighbour

in t

he

dominating set

.

Finally, a total dominating

set D is called a paired dominating set if the

subgraph induced by D has a perfect matching.

1.3 Variations on the proble m

Two variations of the original watchman's walk problem ar

e considered in t

he present

thesis.

Both are motivated by supp

osing that multiple guards are availab

le to monitor

a network.

vVhen determining routes for

mult iple guards on a single graph, a balance

is sought between security and economy:

we want to minimize b

oth t

he t

ime for which

vertices

are

unobserved

as

well

as

the numb

er of guards

we must hire, but the two

are

negatively correlated

. In

the second half of

Chap

t

er 2 we summarize the results

of Hartnell

and Whitehead's Downsizing a dominating set [6],

where t he priority is

given to economy -

they

assume a

fixed number of guards and

attempt to

monitor

the graph as efficiently as

possible wit h

those guards.

In

Chapter 3 we consider

the opposite problem, expanding on a variation first int

roduced by Davies, Finbow,

Hartnell,

Li

and Schmeisser in

[1]. Here

we assum

that a museum cares less about

how many guards are employed than about

protecting

its

valuables . The museum may requir

e, for example, t

hat each room must be

seen every 10 minutes. The goal is to res

pect

t

his t

ime r straint while

using as few guards

as

possible.

We will

say a

vertex is

unobserved if neither t

he vertex nor

any of its neighbours is

occupi

ed by a guard.

Hence for

a given graph G and length of t

ime t,

we are

interested in findin

g

the minimum numb

er of guards

need

ed to

dominate the graph

such t

hat no vertex

is

unobserved for more t han

^t consecut

ive units of

time. More

formally, for fixed time

t E

N ,

a graph G can be t-monitored by a set S of guards

if there exist

s a

function f : S

x N ----.

V

(G) such t

hat

(i)

For

every guard g E Sand at every timeT E N, f(g, ^T

+ 1)

E N[f (g,

r)],

and

(ii)

For

every

vertex v

^E V( G) and every interval I C N of length ^t

+

1, t

here exists

a

guard

^g E

S

and a timeT E I such that f(g,

r )

E

N[v].

Not

e that

f

(g,

r)

is

the ver

tex occupied by

the guard g

at t

ime r . Essentially, condi-

tion (i)

ensures that at each unit of time, guards may only

move fr

om a

vertex t

o one

of its neighbours (i.

e., no 'jumping' is allowed), and condit

ion

(ii) ensures that every

vertex has a guard within

its closed n ighbourhood at least once every t

+ 1 units of time. For

a given graph

G

and length of time

t

, denote by

vllt(G)

the minimum valu

of IS I ,

the number of guards

needed

to

t- monitor a

graph.

In [1], the authors

find

upper

bounds on Wt

(T) for t :::;

3 when

T is a

tree. The

primary obj

ective of the present thesis is

to

generalize

the results

of

[1] by

finding an upper bound on Wt(T) fort

> 3. An upper bound that

holds for all odd

natural

numbers

^t is

presented in Chapter 3. This is followed

by an analysis of the bound, including a

d

escription of a

family of trees for

which it

is

attain d.

In Chapter

4 we

prove bounds for

small even values oft

(t =

2 and

t = 4). Finally, in

Chapter 5 we

discuss a conjectured

upp

r bound for

all even values of t and suggest other

fu

ture

directions for this research.

Chapter 2

Fixed number of guards

In this chapter we review the original watchman's walk problem as well as the 'down-

sizing'

variation. Both of these problems

consider

optimal methods of monitoring

a graph given a

fix d number of guards. 'vVe begin with a singl

guard, the results for which are

primarily from [4].

2.1 One guard: the original watchman proble m

Recall that w1 (G) is the length

of a

minimum

closed

dominating

walk (M CDW)

in

a connected graph G.

Questions of

complexity are among the

first

considered

for

gr aph

theory problems like the watchman's walk; Hartnell, Rall

and

Whitehead

[4]

show

that finding

a

MCDW is NP-complete for

general graphs.

The proof involves relating the watchman's walk problem to

the

well-known Hamilton cycle problem:

given a graph

G, does there

exist a

Hamilton cycle in G? This problem is famously NP-complete [3],

and

we will

see

how it can be used

to show

the same is true

of the

watchman's walk problem. Let CLOSED DoMINATING WALK be phrased

as

follows:

. - - - -- - - - --

given a graph G and positive integer k, is w₁ (G)

<

k? Then we have th following result.

Theorem 2.1.

[4]

^CLOSED DOMINATING WALK is NP-complete.

Proof. Note firstly that CLOSED DOMINATING WALK is in NP, since it is straightfor-

ward to verify any solution to the problem. Given a graph

G

of order n, take

k =

n in the decision problem and create a new graph

G'

by attaching a degree-one vertex to every vertex of G. A MCDW in G' need not visit any of these degree-one vertices, but must visit their neighbours in order to monitor all vertices. Thus every vertex in

G

must be included in a MCDW of

G';

we can conclude that w_{1 (}

G') :::::: k ,

since there are

k

vertices in

G.

If

G

is a Hamiltonian graph then there exists a closed walk of length

k

containing every vertex of the graph, and in this case

w

1 (

G') = k.

H nc if w₁(G')

> k

then

G

is not Hamiltonian, and if we could find a MCDW in

G'

of length k then w could find a Hamilton cycle in the arbitrary graph G, a problem we know

to be NP-complete. 0

We will see that despite the level of complexity for general graphs, there are many types of graphs for which the watchman's walk problem is very approachable. Indeed, the following two lemmas will completely solve the problem for trees.

Lemma 2.2.

[4]

Every cut vertex of a graph G must belong to every dominating walk of G.

Proof. Let v be a cut vertex of G and let W be any dominating walk of G. If HI is the trivial walk on the single vertex v then we are done; otherwise let G1 be a component of

G \

v that contains a vertex of

vV

and let G2 be a second component of

G \ v. If HI

does not pass through

v

then it does not reach vertices of G

_2,

as

v

is t he only vertex in

G

connecting those components.

If u

is a vertex in G

2

then

u

is not on

W

and consequently must b e adj acent to a vertex on the walk. So

u

is adjacent to a vertex of G

1 ;

but then G

1

and G

2

are not separate components of

G \

v, which is a contradiction. Hence every cut vertex belongs to every dominating walk of

G,

as

claimed. D

Lemma 2.3.

[4]

Let G be a connected graph of order at least 3. If W is a MCDW in G then HI does not include any vertices of degree 1.

Proof.

To reach a vertex v of degree 1 the walk must first visit the single neighb our of

v,

from which

it

can dominate

v;

it is therefore unnecessary to add the two extra

edges required to visit v

itself.

D

Not

in

particular that a MCDW in a tree does not include any

leaves. As sug-

gest ed

,

the two preceding

lemmas

tell us exactly how to find a MCDW for any tree.

Since every non-leaf vertex of a tree is a cut vertex, we know the vertex set of any MCDW in a tree will include all non-leaves a nd no leaves; i. e., the vertex set is always

V(T) \ L(T),

for a tree

T.

Since a MCDW must return to t he vertex it starts on, and since t

here is only one path

between any two vertices of a t ree, it

is easy

to see that every edge traversed by a closed walk will in fact b e traversed twice when the graph

is

a t ree. Recall t

hat T0

is the tree

T \ L(T);

then we

have shown w₁(T)

:=:: 2IE(To)l.

Let us find a dominating walk in T.

If

we double every edge of T

₀

t hen every vertex

has even degree and hence there exists an Eulerian circuit in this new tree. Since t he

vertices traversed are all t

he

non-leaves ofT , this circuit

is a closed dominating

walk

ofT

of length 2IE(To)l. A MCDW will be at most this

length

, so w

1

(T):::; 2IE(To)l.

We thus have th following theorem.

Theorem 2.4.

[4]

1fT is a tree then w1(T)

= 2IE(To)l,

and an Eulerian circuit in the tree T₀ with doubled edges is a MCDW forT.

Theorem 2.5.

[4]

For a connected graph G and any spanning tree T of G,

w

₁

(G) ::;

2IE(To)l.

Proof. Let T be any spanning tree of the graph G. We know that a MCDW for T has length

2IE (To)l.

But since

V(G)

=

V (T) ,

this walk is also a closed dominating walk of G, and it follows that a minimum closed dominating walk of G has length

at

most

2IE(To)l.

D

Figure 2.1 illustrat

es

the method described

above

for finding an upp

er

bound on

w

1

(G)

. ote

t

hat the walk obtained is not

a

MCDW,

since

traversing one of the 6-cycles (e.g., the shaded vertices) in

t

his graph gives a shorter closed dominating walk; however

, we can

at least conclude that

w

1

(G) ::;

10.

o :

I I I

0 0 0

Figure 2.1: A closed dominating walk for

G

obtained from the spanning tree

T.

We have already

established

that trees

attain the

upper boun l of Theorem 2.5, since

every non-leaf edge

is traversed twice in a MCDW for

a

tree. Figure

2.2 shows

, - - - - -- - - -

a graph

that is

not a tree that also satisfies w₁

(G) 2IE(To)l,

for the indicated spanning tree

T.

G ZT0 , ,

0 0

_'

o=--= ^--==-=o

'

'0 (j

Figure 2.2: A

graph satisfying

w

1 (G)

= 2IE(T

0 )

I

that is not a tree.

An important

note here is that for a given graph G,

different choices for

the span-

ning

tree T

will

likely result

in

different

values for

IE(To) 1.

Specifically,

a spanning tree with many leaves

will

result

in T

0

having fewer

edges.

Consider again the graph

G in Figure

2.2. Figure 2.3

shows

three

different spanning trees of G and the corre- sponding

closed dominating walks of G

for each. We see that the

upper bound given in Theorem 2.5

can

be

slightly improved if

we

specify

that

the spanning tree

T

be the 'best' spanning tree; i.e.,

that we

choose T so

that

To has the

f west number of

edges. This is equivalent to finding a spanning tree of G with the

maximum number

of leaves, a

problem which

is

known to b

P-hard [3].

The

following

theorem categorizes a class of graphs that do

not

me

t

the bound of Theorem 2.5 for

any choice

of spanning tree

.

Theorem 2.6.

[4]

Let G be a connected graph and letT be any spanning tree of G.

IJG

has girth at least

7

then w₁

(G) < 2IE(To)l.

Proof. Assume

a

graph G has girth at least

7 but that w

₁(G) =

2IE(To)l for

some spanning tree

T of G.

Let u and v

be the

end

vertices

of some edge

in G

that is

not

in

T .

Let

P be the

unique u-v

path

in

T ,

and

let

u' and v'

be the neighbours

. - - - -- - - - -- - - - · - - - -

T T

Figure 2.3: Thr

choices for a spanning tree T , and

the resulting dominating walks.

of ^'U and v, resp

ectively, on P.

Since every vertex on

P

has degree at least two in

T

(except possibly u

a

nd v), this path is contained in

To .

In particular, u' and v' are in

To.

Note that dr₀ ( u', v')

>

3, since otherwise such

a

path from u' to v' together with { u', u'u, u, uv, v, vv',

v'}

would form a

cycle

of length 6 in G, which contradicts that the girth of G is at least 7.

Now, double each edge of T₀ and let vll b

e an the

Eulerian circuit in the resulting multigraph. This

circuit has length

2IE(To)l

and

is thus a MCDW ofT. But if we replace one occurrence of the edges of P ~ E(T_{0 )} from u' to v' (of which there are

at

least 4) on

v\l

with the edges u'u, uv, vv', then we obtain a walkinG that is at least one edge shorter than ^lill. This new walk has all vertices of W

and so

is dominating, which contradicts the fact that W is a MCDW. Thus, there is no such

spanning tree

of G; that is

, no

spanning tree T satisfies w1 (G)

=

2IE(T0) I,

as required.

0

The girth requirement cannot be tightened here, as there do

exist graphs of girt

h

six

that attain the bound in Theorem 2.5 (a cycle of length six, for

example)

. Also note that the converse of Theorem 2.6 is not true, as Figure 2.4 demonstrates that the

upper bound is not

attained for every graph of girth less t

han

seven.

We can

see the

graph

G

has w

1 (G)

< 2jE (T

0 )

I for

every spanning

tree

T

because up

to isomorphism

there

is only one such

T, with the corresponding walk having length 6

, and traversing

the 4-cycle in

G gives a shorter closed dominating

walk.

G

0 D

Figure 2.4: A

graph of girth 4 for which w1

(G) < 2jE(To )l

for any spanning t

ree T .

The fo

llowing theorems consider the watchman's walk problem for several ammon

types of graphs.

Theorem 2. 7.

[4]

If G is a connected graph then w₁ (G) =

0

if and only if G has a dominating verte.'E (that is, a dominating set of size 1).

Proof.

This is trivi

al; the watchman need not

move from the dominating v rtex.

D

Theorem 2.8. [

4]

Let G be a complete multipartite graph. If any part is a single vertex then w1

(G)

= 0, and otherwise w1

(G)

= 2.

Proof. If

one part of a complete multipartite graph is a single vertex, then

that vertex

dominat

es

the entire graph and so, by

Theorem 2.7, w1(G) =

0. Ot

herwise, a vertex u

domin

ates

the v rtices

in all

other p

arts except its own, which can be dominated

by one vertex, say

v, from any other part. Since G is com

plet

e, u and v are adjacent and the

closed walk of length 2 between

them is a

MCDW. Thus w

1

(G) = 2 in t

his

case. D

,---~---- - - - · - - - -

Theorem 2.9.

[4]

Let G be a connected bipartite graph with bipartition (A, B), where both

A

and

B

contain at least

2

vertices. Let

A'

denote a minimum subset of

A

that dominates all of

B ,

and let

B'

denote a minimum subset of

B

that dominates all of A. Then

w 1(G)

~ 2(ma.~{IA'I,

IB' I} ).

Proof. Since G is bipartit , no vertex of A dominates any other vertex of A. Likewise for

B.

Hence, if

A"

is the subset of vertices from

A

on a MCDW then

A"

must dominate B and consequently has at least

lA' I

vertices. Similarly, the set of vertices

B"

from

B on

a MCDW must hav size greater than or equal to

IB'I·

Since we must enter and leave each vertex of the larger of the two sets

A "

and

B ",

our MCDW has length at least twice the cardinality of the larger set, which is at least the larger of

A'

and

B' .

0

Theorem 2.10.

[4]

If Cn is a cycle of length n then

ifn ~ 6 if 3 ::::; n

<

6

Proof. If G is a cycle then we have two clear choices for a 'good' closed dominating walk; either we walk the entire way around the cycle, making a walk of length n, or we

walk partially around the cycle in one direction befor r v rsing and returning to the starting vertex. With any other walk there will be edges traversed more than twice, which adds unnecessary length. Label the vertices of Cn as v1, v2, ... , Vn· Beginning at v3 and walking to Vn ensures every vertex is observed, since the guard can see v₁ from Vn and v₂ from

v

3 . Reversing direction at Vn and returning to 'V3 creates a closed walk that is minimal in the sense that if we had reversed at any vertex befor

Vn then the walk would not be dominating (v1 would be unobserved). This method

.---~~- -

gives a walk of length 2(n ~ 3), which will be shorter than a complete traversal of the cycle if 2(n ~ 3)

<

n, or n

<

6. Hence, w₁

(Cn) =

2(n ~ 3) for n:::; 6, and otherwise

wl(Cn) = n .

D

Notice that in the proof of Theorem 2.10 we are given, in addition to

w

₁

(Cn),

precise constructions for MCDWs in n-cycles. Results about w₁

(G)

and MCD\,Y constructions are known for other families of graphs

G.

Given two graphs

G

and

H ,

the Cartesian product graph

GDH

is the graph with vertex set

V(G )

x

V(H)

and edge set

{(u,v)(u ,v')iu

E

V(G) , vv'

E

E(H)}

U

{ (u ,v)(u'v) lv

E

V(H) , uu'

E

E(G)}.

In [5], for

T

a tree, sharp bounds are found for w₁

(TDKn) ,

and nee ssary and sufficient conditions are found for a walk in TDK2 to be a MCDW. In [4], the following theorem describes MCDWs in cactus graphs.

Theorem 2.11.

[4]

Let G be a connected cactus. Let G' be the induced subgraph of G obtained by deleting all vertices of degree 1, all vertices of degree 2 that are on 3- cycles, and exactly one pair of adjacent vertices of degree 2 from each cycle of length

4

^{or 5 that} contains such a pair of vertices. If each wt edge of G' is duplicated to form

G",

then

G"

is Eulerian and any Eulerian cirwit in

G"

is a MCDW of

G.

Proof. Let G be a cactus graph. W will show that an Eulerian circuit formed as described above is a MCDW by showing that none of the identified vertices need to be on a dominating walk, that each of the remaining vertices (those in G') must be on a dominating walk, and that every edge of

G'

must be traversed in order to connect its v rtices. Doubling th cut edges follows necessarily to ensure that the walk is closed.

By Lemma 2.3, no vertex of degree 1 needs to be on a MCDW, so we eli card

such vertices (i. e., we do not

include t

hem in G' ). Any vertex of degree 3 or higher

,

or of d egr 2 and not on a cycle,

is a cut

vertex in a cactus. To see

this, note that

since every b

lock is a cycle

or an edge, a ver tex belongs t o single block if and only if it

is

on a cycle a nd

has

degree two; otherwise the ver tex belongs to two blocks and its

removal

would disconnect t hose blocks. Cut vert ices must be on

any

dominating walk, by Lemma 2.2, so we keep these vertices in G'.

V ert

ices

of degree 2 on a 3-cycle will be seen fr om the cut vertex (or ve rt ices) on th e cycle, so discard them.

If

t here are adjacent ver tices of degree 2 on a 4-cycle then we discard one pair of them and keep the two remaining a dj acent vert ices, from which a guard can monit or the discarded p air. The edge between t he retained ver tices must be on a dominating walk

in

order for the guar d to m ove from one v r t x to t he other.

If

t

here are no adj

acent vertices of degree 2 on a 4-cycle t hen there are vert ices of degree 3 or h

igher

(i.e., cut vertices) at opp osite corners, which must be on

a

domin at

ing walk. For

a guard t o move between

t

hese opposite ver

tices,

two adjacent edges must b e traversed , and if the guard 's walk is t o b e closed the n two dg s will hav to b e traversed

in the

opposite direct

ion

as well; we can t herefore put all four vertices and all four edges of the cycle in G' . A similar rule applies for 5-cycles.

For cycles of

length 6

or more, a complete tr aver sal suffices for th minimum domin ating walk, even if that t raversal is interrupt ed (at cut vert

ices), and so

G' will

include

full cycles of any

length higher than

5.

Now, any vertex

in

G'

is eit

her a cut vertex or is on a cycle and has degree 2, an l

each cut vertex

is either the end

of a cut edge or only belongs t o cycles.

If a

vertex

only b elongs to cycles then each of it s cycles contr

ibutes

2 incident edges and so t he

total degree of the vertex

is even. Thus,

if we duplicat e each cut edge of G' t o create

G",

all

vertices

of the result

ing graph have even degree. We see that to dominate the

cycles

of G we

can walk each edge

of G',

and

to

connect these

cycl s with

a

closed walk

each cut edge must be walked twice. Such a walk

is obtained pr cisely by finding

an Eulerian circuit in G".

0

Figure 2.5 below demonstrates how Theorem 2.11

applies to t

he given cactus graph G.

G G'

Figure 2.5: A

cactus

G, the graphs G' and G" from Theorem 2.11, and a MCDW.

2.2 Multiple guards : downsizing a dominating set

The original watchman's walk probl m

can

be viewed

as an attempt to minimize t

he unobserved time of vertices, given

a single guard.

In [6]

t

his problem is generaliz

d to

multiple guards, but the question is still

essentially

the same:

given

a fixed

number

of guards, how can we minimize the length of time for which vertices are unobserv

d?

The precise problem addressed in

[6]

i motivated as follows.

Suppose firstly that

a

museum or other network has enough

guards to

place one

at each vertex of a dominating s

t, so that

all ver tices ar e under constant

monitoring.

Let D b

e a

dominating set. If we hav IDI

guards a

nd

ach remains st ationary at a

vertex of D, then we have an

extr mely efficient but expensive security

network.

Now sur pos t

hat the guards hav- b- n lownsized,

so that only some

fraction q,

0

<

^q

<

1, of the guards are now employ d. The following qu

esti n aris s

:

given

qjDI

guards, how can we minimize th maximum time for which any v r tex is unobserv

d?

Given

a closed d ominating walk

in a gr

aph and mult

iple guard

at our disposal, a

natural strategy is to h

ave

the guards '

share' t

he domin

ating

walk, by spacing them out along it a· equally as possible. This will not

always be

the most effective method,

as illustrated in Figure 2.6:

if two guards share the closed

dominating walk on t

he left, which has length 12, then the leaves of this tree are unobserved for

5 cons cutive units

of time, whereas with the two disjoint walks on the right no vert x i unob

erved for

more than 3 units of time. The inefficiency is even more marked when we note that the closed dominating walk in this cas is actually minimum. However, the method

of sharing a

dominating walk

at least gives

us an upper bound on the length of

time

for which vertices must be unobserved

.

Lemma 2.12 formali

zes this

id

a, which i

used r

epeatedly

in [ 6].

Lemma 2.12. If a graph has a closed dominating walk of l ngth m then it can be dominated with p guards such that no veTtex is unobserved fo1· more than

l ^~ l -

¹

units of time.

Figure 2.6:

Different methods

of monitoring a grap

h with two guards (g₁ and g_{2 ).}

Proof. If p guards are spaced out as evenly as

possible along

a

closed walk

of length

m

,

then any two

guards

will be

at

most I~ l

edges apart. If the guards

follow

one another along the walk then every

vertex

on the

walk is occupied

at least once very

I ~ l units of time. Since

the

walk is dominating, this means

every vertex in t

he graph is observed (perhaps from

a

neighbour)

at least once every

I~ l

^units of time,

or

equivalently no vert x is unobserved for more than

I~ l -

¹

units of time.

D

The bound

given in Lemma 2.12

would

obviously

be str

engthened if

the

closed

dominating walk was of minimum

length,

but

since

finding a MCDW in

a general graph is computationally difficult,

we

settle

for

a

cleverly

constructed closed walk

whose vertices

contain

a

given

dominating

set D.

This

construction is outlined

in Theorem 2.14; first

we need

the following

lemma.

Lemma 2.13. If D is a dominating set in a connected graph G then for any set of vertices S <;;;; D there exists a vertex v E D \ S such that de ( v, S) :::; 3.

Proof.

Suppose there

exists a subset S

of

D for which every vertex v

in

D \ S

has

dc(v , S)

~

4. Let v be

any

vertex

in

D but not in

Sand choose

u to b

e the closest

vertex in S to v.

Let

P = u ,

v₁, v_{2 ,}

v

₃, v4, . . . ,

v

be a shortest

u- v

path in

G. The

vertex

v2 is

not in S nor

adjacent to any

vertex in S because

oth rwise u is not t

he

close t vertex to v

inS. Furthermore, v2 and its neighbours are not in D \ S, because

- - - -

these vertic

s

are within distance 3 of

u E S and by assumption the set S

is

at least

distance 4 from

any

vertex

in D \ S.

But then

v2 is not in D and is not adjacent to

a vertex of D , which contradicts t

he fact that Dis a

dominating

set in

G.

^D Theorem 2.14.

[6]

If G is a connected graph with dominating set D then G can be monitor-ed with

qJDI

guards,

0 <

q

<

1, such that no vertex is unobserved for- more than

j% l -

¹ units of time.

Proof. Let v

be

any

vertex

in D.

Construct a

subtree T of G containing th vert

ices

of D

via t

he following iterative procedure. Set v1

=

v, V(Gl)

={vi} ,

E(Gt) =

0, S

₁ =

{vi} ,

and fori

from 2 to I

DI,

find Vi in

D \ si-1

with minimum

dc(vi, si-1)·

Let

pi

be

a shortest path

from Vi to

si-1; by

Lemma 2.13, this path

has length at most 3.

At

each step

the gra

ph

Gi

is connected

b

cause

we

are adding a

path Pi which h

as

one end

already in

the grap

h. Take

T to be a spanning

tree

of t

he final graph ^GIDI·

Note that

at each

step

we add a vertex of

D

and at

most 3

edges to

Gi. Since there

are JDI

-

1 iterations, this shows the graph CIDI (and consequently the tre T) has at most

3(JDI -

1) edges.

Note

also that V(T)

=

V( GIDI) contains every

vertex of D. Hence

if we double the edges ofT we obtain a

clos d dominating walk

of G of length

at most 6(JD

J-1).

Then

by Lemma 2.12 we know qJ

D

I guards can dom

inate

G leaving no

vertex

unobserved for more than

r

6(IDI - 1)1- 1 =

r~- ^{_ 6}

^{1 - 1::;}

r~q1 - ¹

qJDI

q

qJD

I

units

of time, as claimed.

^D

Let us focus

now on

q = ~; i.e., suppose

the

set of JDI guards

have

been cut

by

half. We have

the following

result as an immediate corollary of Theorem 2.14.

Corollary 2.15. If G is a connected graph with dominating set D then

l~ l

guards can monitor G such that no ver-tex is unobser-ved for- more than 11 units of time.

We will see that 1~1 guards are ev n more effective if the vertices of D are suffi- ciently 'close' to one another; i.e., if we have a stronger condition than that guaranteed by Lemma 2.13. In this case we abandon the method of sharing a dominating walk.

The following theor ms explain how we can form clusters of the vertices of D and as- sign a number of guards to each, thereby reducing the total number of edges traversed (see Figure 2.6, for example). We need the following auxiliary graph.

Definition 2.16. Let G be a connected graph with dominating set D. For- a positive integer- d, define G D,d to be the gmph with ver-tex set D in which two ver-tices u, v E D ar-e adjacent if and only if de(u, v) :=:; d.

Theorem 2.17. [

6]

Let G be a connected gmph with dominating set D.

(i) If de ( v, D \ {

v})

:=:; 2 for all v in D then 1 ~I guar-ds can monitor- G such that no vertex is unobser-ved for- mor-e than 7 units of time.

(ii) If de(v, D \ {

v})

:=:; 1 for- all v in D then 1~1 guar-ds can monitor- G such that no ver-tex is unobserved for more than 3 units of time.

Pmof. (i) Assume de( v, D \ { v}) :=:; 2 for all v in D; then by definition the graph G D,2

will have no isolates. Let M be a maximum matching in Gn,_{2 .} All neighbours of an unmatched vertex are end vertices of an edge in Jl;f, since if two unmatched vertices are adjacent then their shared dge could belong to M, which contradicts the fact that M is maximum. For each unmatched vertex we can therefore select an edge incident with a matched neighbour. Now consid r an edge ·u, v of NI. If both u and v are incident with a selected edge then we have a path of length three, say P =

u' , u, v, v',

. - - - -- - - - - -- - - -^{~--- --- -}

in

G

n,₂ where u' and v' are not incident with edges of

NI .

Then

NI

could include the edges u'u and vv' instead of uv, again contradicting its maximality. Thus for each edge of M, exactly one end vertex is now connected to one or more unmatched vertices, thereby creating a collection of stars in G n,2 containing all vertic s of D. The edges in these stars represent paths of length at most 2 in G between two vertices of D.

For each star on ^T vertices, double the edges on the corresponding paths in G and have l~J guards walk an Eulerian circuit in the resulting graph. There are ^{T -} 1 such paths, and when doubled each has length at most 4, so the guards follow each other along the circuit, spaced apart such that no vertex dominated by the walk is unobserved for more than

- 1< - 1= 7

r ^{4(T- lT/2J 1)1 - ~}

^(T-

^{4 (T- 1)/ 1 ) 2 l}

units of time. Since ^T is the number of vertices in each star of

G

n,2 and since these stars compris all vertices of D, placing l~J guards on each star in total uses at most

1~1 guards.

( ii) If de (v, D \ { v}) ::; 1 for all v in D then G ^{D ,}₁ has no isolates and we can form stars in this graph as described above. Each edge in a star corresponds to a single edge in G, so a star on ^T vertices shared by

l

^~

J

guards will have any two guards at most

J2(T - 1) l ^< r ^{2(T - 1)}

1 ^{= 4}

I lT/2J - (T - 1)/2

edges apart. Hence in this case l~l guards can monitor G such that no vertex is

unobserved for more than 3 units of time. ⁰

Note that this method of assigning guards to stars of the graph

Gn , d

can also be

used if D satisfies only de ( v, D \ { v}) :::; 3 for all v in D (using graph G

n ,

3 ); however, in general there is no improvement in this case over the method of sharing a dominating walk. In particular, one finds only that no vertex is unobserved for mor than 11 units of time, which we already have from Corollary 2.15. However, the authors of [6]

note that when many stars created in Theorem 2.17 have odd order r-, the number of guards used in total is actually significantly less than 1~1, since we reduce the number of guards on each odd star from r to l~J =

r2l

^{(recall that we} initially assume every vertex of D has a guard, and that we downsize this set of guards by half). In these cases we can afford to 'waste' guards in certain parts of the graph, while still using only ¹~¹ in total. In particular, if in Theorem 2.17 we eliminate the condition that de(v, D \ { v}) :::; 2 or de(v, D \ { v}) :::; 1 or all v in D, then the resulting isolates in G n,₂ or G D,l could be given their own guard provided there are at least as many odd stars as there are isolates. This gives the following corollary.

Corollary 2.18. [6] Let G be a connected graph with dominating set D. Form a collection of stars in the graph G n,₂ as descr-ibed in the pr-oof of Theorem 2.17; if the number- of odd stars is at least the number of isolates in G n,₂ then

1~1

guards can monitor- G such that no ver-te:r; is unobserved for- more than 7 units of time. If de( v, D \ {

v}) :::;

2 for all v E D and the number- of odd star-s in G

n,

1 is at least the number of isolates in G D,l then 1~1 guards can monitor- G such that no vertex is unobserved for more than 3 'units of time.

Returning to Theorem 2.17, note that when D satisfies de(v, D \ { v}) :::; 1 for all v E D, D is a total dominating set. If the matching NJ defined in th proof of Th or m 2.17 is p rfect th n

D

is in fact a paired dominating set. The following

theorem

shows how paired domination

is ideal for minimizing the length of time for which vertices are unobserved.

Theorem 2.19.

[6]

A graph G can be monitored with -y~G) guards and leave no vertex unobserved for more than 1 unit of time if and only if G has a paired dominating set of size

ry( G).

Proof. (

=>)

If G

can

be monitored by 'Y~)

guards such that every vertex is seen at

least once

every two

units of time then the

set S

1 of vertices occupi

d by the guards

at some time

t and t

he set S₂ of vertices occupied at time

t +

1 must together form a domin

ating set of G;

i.e.,

D = sl us2

is a dominating set of

G.

Thus

ID I 2:: ry(

G). But

since there are -y~G) guards,

we must have

I S1I , IS2 I :::;

-y~G),

so ID I :::;

-y(~)

+

-y(~)

=

ry(G)

and consequently I DI =

ry(G). We

conclude that sl n s2 = 0, and if we

let Nf be the

set

of

edges

walked by the guards,

each

having

one end

vertex in

sl and one in s2 ,

then Nf is a perfect matching in the subgraph induced by D, and hence D is

a pair d

dominating set, as required.

( {:::) If G has a paired dominating

set

D

then

the

subgraph

induced by D has

a

perfect matching, M, whose

end

vertices

comprise

D. Each

edg of

Nf can be traversed repeatedly by one

guard, so that

no vertex is unobserved for more than 1 unit of time,

and this

method uses

exactly

1~1

guards. D

- - -- ----- - - -

Chapter 3

Fixed time

In this chapt r we explore a variation on the watchman's walk problem first introduced by Davies et al. in

[1].

This

variation takes t

he opposite standpoint of the problem discussed in

Chapter

2,

assuming

that fixed

time constraints ar

imposed on the monitoring of

a graph G and attempting to determine t

he minimum number of guards, Wt (G), required to meet those

constraints.

We begin with

an

introduction to this problem, including som

e basic results,

and proceed to find

an upper bound on

Wt( G)

for any odd integer t >

0.

3.1 Introductory results

Recall that a graph G

can b

et-monitored by m

guards if t

here exists a collection of m

walks (not

necessarily distinct or

disjoint) t

hat can be

traversed by t

he guards such

t

hat no vertex in G is unobserved for

more t

han

t

units of time. Equivalently,

every

vertex is either occupied by a guard or

adj acent

to

a

vertex

occupied

by

a guard at

least once every t

+

1 units of time.

Suppose for example that the graph

G in

Figure 3.1 below must be dominated such that no vertex is unobserved for more than t = 2 units of time. The gray vertices indicate the positions of four guards

9_{1 ,}

9

_{2 ,} 9₃

and

9₄

at some fixed point in time, and t

he

dotted arrows indicate the direction from which the guards have entered their current vertices. We will see

how

t

hese

four guards can 2-monitor G. The guard 9

1

traverses two edges, and all four of the vertices dominated by 9

1

are seen at least once every 2 units of time. The guard 9

2 remains stationary

at the indicated vert x, thereby constantly dominating that vertex and its two neighbours. Guards

93

and

94

share a single closed walk, the intention being that the guards are spaced equally apart and follow one another along the walk. The reader can v rify that this ensures no vertex dominated by

9₃

and

9₄ is

unobserved for more than t = 2 units of time.

G

Figure 3.1: A graph

G

that is 2-monitored with four guards.

In Figure

3.1, each guard repeatedly traverses a

closed walk. Although

there is

no such stipulation in the

definition

of t-monitoring, we may in fact assume this is

always the case.

Let G

be a graph t-monitored by guards and suppose one or more

of these guards share a walk W that is not closed. At any fixed point in time, label a vertex 0* if it is currently occupied by a guard, label a vertex 0 if it is unoccupied but adjacent to a vertex with a guard, and label every other vertex with a positive integer (at most t) according to the length of time since the vertex was last observed. For example, from the graph G in Figure 3.1 we obtain the vertex labelling shown in Figure 3.2 below. Since both t and

IV(T)I

are finite, there are only finitely many such labellings, and so at some point a vertex labelling will be repeated. When this happens, we can truncate

W

and have it repeat whatev r edge sequence followed the first occurrence of that labelling. The new walk is closed does not disrupt the t-monitoring of

G.

Since any non-closed walk can be reconstructed in thi way, have the following theorem.

G

0 1 0

o"

Figure 3.2: The length of time for which each vertex in G has been unobserved.

Theorem 3.1. If a graph G can bet-monitored by m guards then G can bet-monitored by m guards whose walks are closed.

ote that for any connected graph G, if m guards can (minimally) monitor G such t hat each vertex

is seen within every

t + 1 units of t ime, t hen with those

m guards

each vertex

is a

lso seen within every t + 2 units of time. Thus for any t , we have

Wt (G) ~ Wt+ 1

(G). This

idea is

summari zed

in

Lemma 3. 2 and will be useful as w

invest

igat e increasing values oft.

Lemma 3.2.

For any graph G

₇ T!V₀(G) ~

VV

₁

(G )

~ W₂(G) ~ ....

In [1], the a

uthors discuss bounds on W

t (G) for variou

s values of t, wit h G usually assum d t o b e a tree. vVe b egin naturally with t = 0;

in

this case,

Wt (G)

= I'( G) , since if vertices cannot b e

unobserved for

even a single unit of time then the guards must dominate all vertices while remaining stationary.

In t

his section , when a graph

G is

clear from the context , let

n represent

I V (G)

1.

Theorem 3.3.

[1] For any connected graph G

7

W

0

(G )::::;

l~J.

Proo f. For any dominating set

D

of a conn ct d gr aph G , the set

V(G)\D

is also a dominating set of G.

Hence a

minimum dominating set must have cardinality

less

t han or equ al t o

l~J,

as otherwise its se t complement is a dominating set wit h fewer vertices . Thus I' ( G)::::;

l~J,

and since

W0(G)

=I'( G) , the result foll ows. 0

The trees of even order

n

that have domination number equal to

~

are classified

in [2]: they are composed of

an equal number of

leaves and non-leaves,

wi t h every non-leaf adjacent to exactly one

leaf. Because

of t

he complexity

of t he t ime restraint problem for gen eral graphs, the remainder of this chapter predominantly considers trees.

Theorem 3.4 summarizes three importa

nt results of [

1],

pertaining t o upper bounds

on the values of

W1(T), W2(T),

and

W3(T)

for an arbi t rary t ree

T. In the next section